Bayesian inference

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Bayesian inference

Primer

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Likelihood Posterior

Prior

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⇡(x) ⇡(y |x)

⇡(x |y)

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⇡(x) ⇡(y |x)

⇡(x |y)

Bayes’ theorem

posterior = prior ⇥ likelihood evidence

prior

posterior likelihood

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Bayes’ theorem

Parameter identification: Bayesian approach

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Bayes’ theorem

Descriptive formula

Parameter identification: Bayesian approach

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A discrete example of Bayes’ theorem

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This is our prior information for the probability of each face: 1/6

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P=1/6

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Assume that after throwing the dice, you see the above evidence

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Goal: determine the probability of this evidence for each face of the dice

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a

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b

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c

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d

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a b

c

d

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a b

c

d

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a b

c d

One would never see a dot at the star positions for this face 

The probability of the evidence is zero

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a b

c d

Two possibilities (a,c) and (b,d)

a b

c

d

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Also two possibilities (a,c) and (b,d)

a b

c d

a b

c

d

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a b

c

d

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a b

c

d

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a

b c

d

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a

b c

d

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a b

c d

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a b

c d

a

b c

d

a b

c d

a

b c

d

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a b

c d

a

b c

d

a b

c d

a

b c

d

Four possibilities

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Four possibilities

a b

c

d

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Four possibilities

a b

c

d

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a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

0 2 2

4 4 4

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a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

0 2 2

4 4 4

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Probability that this was the face of the dice knowing

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P=0

P=⅛

P=¼

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P=0 P=⅛ P=¼

P=1/6

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Stress-strain data

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✏

Identify the parameters

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Model

observations=f(parameters, error)

Construct the likelihood function

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Additive noise model

Noise model

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Likelihood function for additive model

Likelihood function

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Constitutive model

Observed data

Constitutive law: linear elasticity

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Prior information on Young’s modulus

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Error model (noise)

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Likelihood function

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Bayes’ theorem: calculate the posterior

posterior = prior ⇥ likelihood

evidence

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⇡(x)

Bayes’ theorem: calculate the posterior

posterior = prior ⇥ likelihood evidence

prior

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⇡(x) ⇡(y |x)

Bayes’ theorem: calculate the posterior

posterior = prior ⇥ likelihood evidence

prior

⇡(y |x) = N(y x✏, 0.0001)

⇡(y |x) = ⇡(!) = ⇡(y f (x))

likelihood

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⇡(x)

⇡(x |y)

Bayes’ theorem: calculate the posterior

posterior = prior ⇥ likelihood evidence

prior

posterior

⇡(y |x) = N(y x✏, 0.0001)

⇡(y |x) = ⇡(!) = ⇡(y f (x))

⇡(y |x)

likelihood

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Posterior probability

⇡(x |y) = R ⇡(x)⇡(y |x)

⌦ ⇡(x)⇡(y |x)dx

⇡(x)

⇡(y |x)

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The 99.73% rule: observations

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Propagation of the uncertainty to the constitutive model

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Perfect plasticity

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Perfect plasticity

Constitutive model

x(2)

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Observed data

Modified form for constitutive model

Perfect plasticity

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Perfect plasticity: contour plot of prior in parameter space

parameter 1 parameter 2

“wider” prior for yield stress -> more info needed

“narrow” prior for E y0

E

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Posterior probability

Perfect plasticity

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Posterior probability

Perfect plasticity

likelihood for each   observation

1/ ²

(x µ) ²

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Posterior probability

Perfect plasticity

likelihood for each   observation

stress model stress measurement

1/ ²

(x µ) ²

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Posterior probability

Perfect plasticity

likelihood for each   observation

stress model stress measurement

1/ ²

f (x |µ, ) = 1

p 2⇡ e ^(x ^{2 2} ^µ)2 (x µ) ²

Difficult to compute the evidence probability: use MCMC

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Markov-Chain Monte Carlo (MCMC) method: parameter space

too small adequate

too large too large

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